The Asymptotic Behaviour of Semigroups of Linear Operators

Jan van Neerven

ERRATA

• Author's address: P.O. Box 5031, 2600 GA Delft.
• Page 6, Proof of Proposition 1.1.5: The spectral mapping formula should read $\sigma(R(\lambda,A))\setminus\{0\} = \frac{1}{\lambda-\sigma(A)}$.
• Page 7, Line one of Proposition 1.1.7: the growth bound $\omega_0({\bf T}_Y)$ is introduced only in the next section. Instead take Re$\lambda > \omega$, where $\omega$ is any real number such that $|| T(t) || \leq Me^{\omega t}$ for some $M \geq 0$ and all $t\geq 0$.
• Page 111, lines 18-20: This has been resolved.
• Page 122, line 1: replace `for all $\a\ge 0$ with $\a>{1\over p}$' by `for all $\a>{1\over p}$'.
• Page 122, Lemma 4.3.1: The estimate for G should read: $||G(\lambda)|| \le M(1+|\lambda|)^{\max\{-\alpha,-1\}}$. The mistake in the proof occurs in the last estimate on p. 123, where the term in the integral corresponding to $(\lambda_0 + t)^{-1}$ is estimated in an incorrect way. I thank Charles Batty for pointing this out to me. As a consequence, Theorem 4.3.2 only holds for $p\in (1,2]$ (by inspection of the proof). Finally proof of Theorem 4.4.2 needs some changes. This can be done along the lines of the argument on p. 141 by adding an extra term of the form $(\omega+is)^{-\beta}$ to ensure $L^1-$integrability. The details will be worked out in the joint paper with Sen-Zhong Huang, `$B-$convexity, ... ', to appear in J. Math. Anal. Appl.
• Page 132: Replace lines 4-8 `if the local resolvent ... for arbitrary $X$ we proved that' by: `if the underlying space $X$ is $B-$convex, and that $$ \limt \n\t (\l_0-A)\up{-\a}x_0\n=0$$ for all $\a>1$ if $X$ has the analytic Radon-Nikodym property; in both cases we assume that the local resolvent $\l\mapsto \r x_0$ extends to a bounded holomorphic function in the open right half-plane. Also, by means of an example we showed that these results fail if no restrictions on $X$ are imposed. Further, for arbitrary $X$ we proved that'
• Page 141, line 3: Replace `${\cal F}g_{\omega,x,x^*}(t) = \langle x^*, T(t)(\lambda_0-A)^{-\alpha-\beta} x \rangle$' by `${1\over 2\pi} {\cal F}g_{\omega,x,x^*}(t) = \langle x^*, T(t)(\omega-A)^{-\beta}(\lambda_0-A)^{-\alpha} x \rangle$'.
• Page 160, line 13: `By (iii),'.
• Page 168, formula (5.3.1): replace `$\n G_s(\l)\n$' by `$\n G_s(z)\n$'.
• Page 171, last formula: replace `$t$' by `$\tau$' in the integrand.
• Page 172, first formula: replace `$t$' by `$\tau$' in the integrand.
• Page 175, Example 5.3.12: replace `$f$' by `$f_k$' in the right hand side of the definition of $f(t)$. The semigroup has to act on $Y:=C_0(\RR_+)$ of course; our argument then shows that the codimension-one subspace $X=C_{00}(\RR_+)$ of $Y$ is contained in the closed linear span of the orbit of $f$. Choosing $t_0>0$ such that $(U(t_0)f)(0)\not=0$, one has $U(t_0)f\not\in X$. It follows that the closed span of the orbit of $f$ is all of $Y$.
• Page 179, line 11: `$A\s x\s = i\o x\s$'.
• Page 210, lines 15-20: Example 5.1.12 is from [BV1] and 5.1.14 is from [BNR1].
• Page 216, equation (A.1.3): `$t^{-\alpha}$'.

TYPOGRAPHICAL ERRORS

• Page 5, line 20: `facts'.
• Page 110, line -1: `He actually proves a slightly more ...'
• Page 141, line 7: replace `4.2.3' by `4.2.2'.
• Page 157, line 1: drop the comma.
• Page 206, line 21: Add `.'
• Page 211, line 9: `For the real line ...'
• Reference [HNR]: Omit first `and'.

Last update: June 29, 1998. If you know mistakes that are not in this list, please send me an e-mail: